Numerical Analysis of Fluvial Landscapes

The Smith and Bretherton model for fluvial landsurfaces consists of a pair of partial differential equations: one governing water flow and one governing sediment flow. Numerical solutions of these equations have been shown to provide realistic models of the evolution of fluvial landscapes. Further analysis of these equations shows that they possess scaling laws (Hack's Law) that are known to exist in nature. The preservation of these scaling laws in simulations is highly dependent on the numerical method used. Two numerical methods, both optimized for overland flow, have been used to simulate these surfaces. The implicit method exhibits the correct scaling laws, but the explicit method fails to do so. These equations, and the resulting models, help to bridge the gap between the deterministic and the stochastic theories of landscape evolution. Slight modifications have been made to this model to make the resulting surfaces more realistic. The most successful of these was the addition of an abrasion term to assist in the channelization of rivers.